I think they are doing their very best to make sure, to not break any savegames. Unfortunately though, not every feature is doable without making that decision and before hampering the development and delaying any content I'm completely fine with breaking savegames.
Either try copying the map_p.bin file from one save to another or there is still the possibility to cheat your char back up if you really want to mitigate the leveling-process.

@Necromancer had the same problem, Enigma told me to delete the media folder of the game and validate it again. TIS changed the names of the texture files for cars and Steam is not properly downloading the updates.

Characters are stored on the client, so you would have to copy the savegame everytime you switch PC. You could also move your Zomboid-Folder to a cloud-folder that gets synced (look here at the cachedir option).

Mostly playing MP with my friends and we usually quit after we get self-sustainable, which differs due to my mood when I set the difficulty. We normally play for two months before we hit this point.
As for why we only play such a "short" amount: I think this is another topic and I will start another thread for it.

So curious about the new climate system and now with the fancy detection of playerbuilt stuff/sealed areas this is really nice and should open up some new possibilities (claiming of playerbuilt structures).

I personally use LinuxGameservermanger (short LGSM).
LGSM is a shell script that has many comfort functions for running a gameserver in linux. I find the setup and configuration pretty straightforward, might want to give it a shot.